What is Predicate Calculus?
There are:
- An infinite supply of individual variables,
- An infinite supply of individual constants.
- Some finite number of predicates each of which takes some non-negative number of arguments each of which is an individual identifier.
An individual identifier is either an individual variable or an individual constant.
A universal quantifier is ‘Π’ followed by an individual variable.
A proposition is either:
- A predicate followed by the right number of individual identifiers for that predicate,
- An implication sign (“C”) followed by two propositions,
- A not sign (“¬”) followed by a proposition,
- A universal quantifier followed by a proposition.
An individual variable is free in a proposition just in case
it occurs freely in that proposition.
An occurrence of an individual variable, x, in a proposition is free if that occurrence is not also an occurrence of a sub-proposition that begins Πx.
A sentence is a proposition with no free variables.
It follows that every proposition is some finite number of symbols and that the sentences may be enumerated.